Selected Research Topics

Quantum Cryptography

• For a popular introduction to quantum cryptography see the CQC tutorial or the May 2005 issue of the +Plus magazine .

• Original idea: In 1990 I began developing my own approach to quantum cryptography based on quantum entanglement. It was both different and independent from the earlier work on quantum key distribution based on the uncertainty principle. In fact, until the early 1990s quantum cryptography was basically unknown. I took my inspiration from the Einstein, Podolsky and Rosen paper , in which the authors discuss the completeness of quantum theory and, among other things, define an element of reality as "...If, without in any way disturbing a system, we can predict with certainty… the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity”. Read it again and you realize that this is also a definition of perfect eavesdropping! I guess I was lucky to read it in this particular way. The rest was just about rephrasing the subject in cryptographic terms. My paper on the entanglement-based quantum key distribution [D.Phil. Thesis, Oxford 1991; Physical Review Letters, 67, pp.661-663, (1991).] generated a spate of new research that established a vigorously active new area of physics and cryptology and is still the most cited paper in quantum cryptography.

• Experiments : My journey to a quantum optics lab in Malvern started on the slopes of Italian Dolomites. This is how I met John Rarity, an experimental physicist working for the Defence Research Agency (DRA) in Malvern (today QinetiQ). It did not take long to persuade him to produce the first entangled photons to carry secret bits. Together with Paul Tapster, John's colleague from DRA, we introduced parametric down-conversion, phase encoding and quantum interferometry into the repertoire of cryptography. It was great fun to be involved in setting up the first experimental demonstration of entanglement-based key distribution [ Physical Review Letters 69, pp.1293-1295 (1992)]. In 1991 experimental quantum cryptography based on quantum entanglement became reality. Einstein’s “spooky action at a distance” found its first practical application.
  
  The picture shows parametric down conversion used for quantum key distribution at the Defence Research Agency in Malvern circa 1991. The intensities of the two down-conversted beams were too low to be visible on a photograph; their location is marked by the two reference beams (red and green).
  
  
• Security Proofs: In the early 1990s there was very little understanding of the relations between noise, stability and security of quantum key distribution. I still recall rather animated discussions on the subject during the Dagstuhl meeting in 1993 (The European Institute for System Security Workshop on Quantum Cryptography and Quantum Information Theory). My attempts, with Bruno Huttner, to clarify the situation resulted in initiating studies of quantum eavesdropping [Journal of Modern Optics, 41 pp.2455-2466 (1994)]. It was a pleasure to join forces with Asher Peres while working on this particular topic [ Physical Review A 50, pp.1047 - 1056 (1994)]. In the summer of 1995, after sampling some good wine from Piedmont ( Grignolino d'Asti :)) in the city of Turin, Richard Jozsa, Sandu Popescu and I started discussing the security issues in noisy quantum channels. Together with our colleagues we proposed quantum privacy amplification, a technique which uses entanglement purification to make the entanglement-based quantum cryptosysems operable and secure even over noisy quantum channels [D.Deutsch et al. Physical Review Letters, 77, pp.2818 - 2821 (1996) or here ]. N.B. Grignolino d'Asti in the acknowledgements in our PRL paper! It turned out to be a very powerful technique; quantum privacy amplification was subsequently used in most security proofs of quantum key distributions. More recently Matthias Christandl, Renato Renner and I have developed a different technique for security proofs; it is based on bounds on the performance of quantum memories (quant-ph/0402131).

Quantum Computation

• Algorithms: A common pattern underpinning all known quantum algorithms can be identified when quantum computation is viewed as...
  • [a physicist's view] ...multi-particle interference. With Richard Cleve, Chiara Macchiavello and Michele Mosca I have developed and used this approach to review and to improve some of the existing quantum algorithms and to show how they are related to different instances of quantum phase estimation [Proceedings of the Royal Society A, 454 pp. 339-354 (1998)]. In passing we have simplified early algorithms of Deutsch and Deutsch-Jozsa, and provided a neat analysis of a quantum Fourier transform.
  • [a mathematician's view] ....searching for hidden subgroups. In collaboration with Michele Mosca I have also provided a unified mathematical picture of quantum algorithms in terms of group theory and described its connection with eigenvalue estimation on a quantum computer. The hidden subgroup problem also encompasses the problem of finding orders of elements in a group, of which the factoring algorithm is a special case [The hidden subgroup problem and eigenvalue estimation on a quantum computer, Lecture Notes in Computer Science Vol.1509, pp 174-188, Springer 1999 (Ed. C.P. Williams)].
• Universality
  I worked on the basic constituents of quantum computers, i.e. quantum logic gates and quantum Boolean networks. In winter 1994 I spent many evenings and nights on fascinating discussions with David Deutsch (David is not an early person) and my first Ph.D. student, Adriano Barenco, which led us to proving that almost any quantum logic gate operating on two quantum bits is universal [Proceeding of the Royal Society A 449, pp. 669-677 (1995).]
• Implementations
  I came up with few early proposals for implementations of quantum computation. Probably the most interesting one is the idea of using the induced dipole-dipole coupling in an optically driven array of quantum dots (shown on the left) [Advances in Quantum Phenomena ed. E.G. Beltrametti and J-M. Levy-Leblond, pp. 243-262, Plenum Press, New York, (1995) see also A.Barenco, D.Deutsch, and R.Jozsa, Physical Review Letters 74 pp.4083-4086 (1995) ]. Although this array of quantum dots may never be used for quantum coherent computation, the idea alone stimulated researchers at the HP Basic Research in Palo Alto (Stan Williams and his colleagues) to design novel molecular processors for more conventional computation.
  
  The picture on the left shows the first experimental proposal for quantum computation with an array of single-electron quantum dots. A dot is “activated” by applying suitable voltage to the two metal wires that cross at that dot. Similarly, several dots may be activated at the same time. Because the states of an active dot are asymmetrically charged, two adjacent active dots are each exposed to an additional electric field that depends on the other’s state. This way the resonant frequency of one dot depends on the state of the other. Light at carefully selected frequencies will selectively excite only adjacent, activated dots that are in certain states. Such conditional quantum dynamics are needed to implement quantum logic gates
  
  Working on implementation of quantum computation gives a wonderful excuse to spend some time learning about fascinating quantum technologies, in my case about surface acoustic waves and photonic crystals .
• Geometric quantum computation: Together with Jonathan Jones, Vlatko Vedral and Giuseppe Castagnoli [Nature 403, pp. 869-871 (2000)] I have proposed the implementation of a conditional Berry phase between two nuclear spins. Combined with one-spin operators, this simple operation is a universal gate for quantum computation; any unitary transformation can be implemented with arbitrary precision using only one-spin operations and conditional phase shifts. Thus quantum geometrical phases can form the basis of any quantum computation. Moreover, as the induced conditional phase depends only on the geometry of the path executed by one of the spins it is resilient to certain types of errors and offers the potential of a naturally fault-tolerant way of performing quantum computation. I have also looked into a problem of operational definitions of geometric phases for mixed states, both from the theory side [ Phys.Rev.Lett. vol.85 pp.2845-2849 (2000)] and experimental realisations [ Phys.Rev.Lett. 91, 100403 (2003)].

Decoherence Free Subspaces

Miscellaneous

• Entanglement swapping
  I have proposed several novel uses of quantum entanglement, in particular the process of "entanglement swapping'' [with A.Zeilinger, M. Zukowski , and M.Horne, Physical Review Letters, 71, pp.4287-4290 (1993)] which opens possibility of long distance quantum communication using quantum repeaters.
• Better atomic clocks
  I have contributed to the improvement of quantum frequency standards and atomic clocks via prescribed entangled states [with S. Huelga et al., Physical Review Letters 79 pp. 3865-3868 (1997)].
• Optimal state estimation
  With Rado Derka and Vladimir .Buzek, I constructed a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system [Physical Review Letters, 80 pp. 1571-1575 (1998)].
• Quantum cloning
  I got also interested in a related issue of optimal quantum cloning. Working with Dagmar Bruß and Chiara Macchiavello on the best possible approximation to a universal quantum cloning machine, and introducing the “black cow factor” was lots of fun [Physical Review Letters, 81 pp.2598-2601 (1998)]. Our result was general and applied to cloners which operate on mixed input states.
• Quantum purification
  Ignacio Cirac, Chiara Macchiavello and I proposed a multi-qubit measurement which projects on spaces of partial symmetries under the exchange of qubits, labeled by the Young tableaux [Physical Review Letters 82, pp.4344–4347 (1999)]. Apart from the optimal qubit purification this measurement has many interesting applications, in particular in the estimation of spectra of density operators.
• Quantum state transfer
  Quantum computers will need quantum wires, e.g. chains of qubits that admit the perfect state transfer of any quantum state in a fixed period of time. Nilanjana Datta, Claudio Albanese, Matthias Christandl, Alastair Kay, Andrew Landahl and I showed how a perfect state transfer can be achieved over arbitrarily long distances in a linear chain of coupled spins. Our construction is especially appealing as it requires no dynamical control over individual inter-qubit interactions [ Phys.Rev.Lett. 92, 187902 (2004), Phys.Rev.Lett. 93, 230502 (2004), and Phys.Rev. A71, 032312 (2005)].

Mathematical truth and physics